Quantum error correction involves protecting the information in say, a \(d_0\) -dimensional Hilbert space \(\mathcal {H}_0\) , by encoding it in a \(d_0\) -dimensional subspace \(\mathcal {C}\)  of a larger Hilbert space \(\mathcal {H}\) . Subsequently, a recovery map \(\mathcal {R}\)  is applied in order to reverse the effects of errors, followed by decoding, which brings the information back into the original Hilbert space \(\mathcal {H}_0\) . The encoding \(\mathcal {W}\)  is an isometry whose action can be described as, \(\mathcal {W}:\mathcal {B}(\mathcal {H}_0)\rightarrow \mathcal {B}(\mathcal {C})\subseteq \mathcal {B}(\mathcal {H})\) , where \(\mathcal {B}(.)\)  denotes the set of bounded linear operators on the respective Hilbert spaces. The subspace \(\mathcal {C}\)  is called the codespace or simply, a quantum code [1]}, [2]}.
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